Question: Simplify the following expression: $y = \dfrac{7x^2- 12x- 4}{7x + 2}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-4)} &=& -28 \\ {a} + {b} &=& &=& {-12} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-28$ and add them together. Remember, since $-28$ is negative, one of the factors must be negative. The factors that add up to ${-12}$ will be your ${a}$ and ${b}$ When ${a}$ is ${2}$ and ${b}$ is ${-14}$ $ \begin{eqnarray} {ab} &=& ({2})({-14}) &=& -28 \\ {a} + {b} &=& {2} + {-14} &=& -12 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({7}x^2 +{2}x) + ({-14}x {-4}) $ Factor out the common factors: $ x(7x + 2) - 2(7x + 2)$ Now factor out $(7x + 2)$ $ (7x + 2)(x - 2)$ The original expression can therefore be written: $ \dfrac{(7x + 2)(x - 2)}{7x + 2}$ We are dividing by $7x + 2$ , so $7x + 2 \neq 0$ Therefore, $x \neq -\frac{2}{7}$ This leaves us with $x - 2; x \neq -\frac{2}{7}$.